Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}-4x+6y &= -2 \\ -6x-2y &= -4\end{align*}$
Begin by moving the $y$ -term in the second equation to the right side of the equation. $-6x = 2y-4$ Divide both sides by $-6$ to isolate $x$ $x = {-\dfrac{1}{3}y + \dfrac{2}{3}}$ Substitute this expression for $x$ in the first equation. $-4({-\dfrac{1}{3}y + \dfrac{2}{3}}) + 6y = -2$ $\dfrac{4}{3}y - \dfrac{8}{3} + 6y = -2$ Simplify by combining terms, then solve for $y$ $\dfrac{22}{3}y - \dfrac{8}{3} = -2$ $\dfrac{22}{3}y = \dfrac{2}{3}$ $y = \dfrac{1}{11}$ Substitute $\dfrac{1}{11}$ for $y$ in the top equation. $-4x+6( \dfrac{1}{11}) = -2$ $-4x+\dfrac{6}{11} = -2$ $-4x = -\dfrac{28}{11}$ $x = \dfrac{7}{11}$ The solution is $\enspace x = \dfrac{7}{11}, \enspace y = \dfrac{1}{11}$.